Toric geometry and mirror symmetry

by Mavlyutov, Anvar Renatovich

Abstract (Summary)

In this dissertation, we first study complete intersections of hypersurfaces in toric varieties. We introduce a quasismooth intersection in a complete simplicial toric variety, which generalizes a nonsingular complete intersection in a projective space. Using a Cayley trick, we show how to relate cohomology of a quasismooth inter section to cohomology of a quasismooth hypersurface in a higher dimensional toric variety. The cohomology of quasismooth intersections of ample hypersurfaces is completely described. Next, we study semiample hypersurfaces in toric varieties. While the geometry and cohomology of ample hypersurfaces in toric varieties have been studied, not much attention has been paid to semiample hypersurfaces de fined by the sections of line bundles generated by global sections. It turns out that mirror symmetric hypersurfaces in the Batyrev mirror construction are semiample, but often not ample. We show that semiample hypersurfaces lead to a geometric construction which allows us to study the intersection theory and cohomology of the hypersurfaces. The toric Nakai criterion is proved for ample divisors on complete toric varieties. A similar result shows: the notions nef (numerically effective) and semiample are equivalent. Then we study the middle cohomology of a quasismooth hypersurface. There is a natural map from a graded (Jacobian ) ring to the middle cohomology of the hypersurface such that the multiplicative structure on the ring is compatible with the topological cup product. Finally, we study the chiral ring of Calabi-Yau hypersurfaces in Batyrev's mirror construction, widely used in physics and mathematics. This ring is important in physics because it gives the correlation functions describing interactions between strings. From a mathematical point of view, this also produces enumerative information on mirror manifolds (e.g., the number of curves of a given degree and genus). We show that for a quasismooth hypersurface there is a ring homomorphism from its Jacobian ring to the chiral ring. In the Calabi-Yau case, we get an injective ring homomorphism from a quotient of the Jacobian ring into the chiral ring. We construct new elements in the chiral ring, which should correspond to non-polynomial deformations (moving the Calabi-Yau outside the toric variety). The main result is an explicit description of a subring of the chiral ring of semiample regular Calabi-Yau hypersurfaces. This contains all information about the correlation functions used by physicists. Computation of the chiral ring leads to a description of cohomology of the hypersurface. In particular, we describe the toric part of cohomology of a semiample regular hypersurface defined as the image of cohomology of the toric variety.